In a saucepan, stir 3 cups cold water into the mung bean powder.
Set the pan over medium-high heat and cook, stirring, until the mixture thickens and becomes nearly translucent, 3 or 4 minutes. Pour the mixture into a loaf pan. Let cool at room temperature until completely solidified, about 1 hour.
Heat 1 tablespoon of the oil in a nonstick skillet over medium-high heat. Add the egg, tilting the pan so the egg covers the surface in as thin a layer as possible. Cook until it’s just set, about 1 minute, then carefully flip the omelet over and cook for another minute. Slip the omelet onto a plate and let it cool.
Add the remaining 1 tablespoon oil to the same pan and heat over medium-high heat. Add the mushrooms and cook, stirring now and then, until browned and softened, about 6 minutes. Set aside. Slice the cooled egg omelet into thin ribbons and set aside.
Turn the mung bean jelly out of the loaf pan onto a cutting board. Cut the jelly crosswise into thin slices as if it were a loaf of bread, then cut each slice lengthwise into noodles. In a large bowl, whisk together the soy sauce, vinegar, red pepper powder, scallions, and garlic. Add the mung bean noodles, cucumber, radishes, ginger, and sprouts to the bowl and toss gently to combine. Garnish the noodles with the reserved egg ribbons and shiitakes.